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Rational Functions and Models
Lesson 4.6
Definition
Consider a function which is the quotient of two polynomials
Example:
( )( )
( )
P xR x
Q x Both polynomials
2500 2( )
xr x
x
Long Run Behavior
Given
The long run (end) behavior is determined by the quotient of the leading terms Leading term dominates for
large values of x for polynomial Leading terms dominate for
the quotient for extreme x
11 1 0
11 1 0
...( )
...
n nn nm m
m m
a x a x a x aR x
b x b x b x b
nnm
m
a x
b x
Example Given
Graph on calculator Set window for -100 < x < 100, -5 < y < 5
2
2
3 8( )
5 2 1
x xr x
x x
Example
Note the value for a large x
How does this relate to the leading terms?
2
2
3
5
x
x
Try This One
Consider
Which terms dominate as x gets large
What happens to as x gets large?
Note: Degree of denominator > degree numerator Previous example they were equal
2
5( )
2 6
xr x
x
2
5
2
x
x
When Numerator Has Larger Degree
Try
As x gets large, r(x) also gets large
But it is asymptotic to the line
22 6( )
5
xr x
x
2
5y x
Summarize
Given a rational function with leading terms
When m = n Horizontal asymptote at
When m > n Horizontal asymptote at 0
When n – m = 1 Diagonal asymptote
nnm
m
a x
b x
a
b
ay x
b
Vertical Asymptotes
A vertical asymptote happens when the function R(x) is not defined This happens when the
denominator is zero Thus we look for the roots of the denominator
Where does this happen for r(x)?
( )( )
( )
P xR x
Q x
2
2
9( )
5 6
xr x
x x
Vertical Asymptotes
Finding the roots ofthe denominator
View the graphto verify
2 5 6 0
( 6)( 1) 0
6 or 1
x x
x x
x x
2
2
9( )
5 6
xr x
x x
Zeros of Rational Functions We know that
So we look for the zeros of P(x), the numerator
Consider
What are the roots of the numerator? Graph the function to double check
( )( ) 0 ( ) 0
( )
P xR x P x
Q x
2
2
9( )
5 6
xr x
x x
Zeros of Rational Functions
Note the zeros of thefunction whengraphed
r(x) = 0 whenx = ± 3
Summary
The zeros of r(x) arewhere the numeratorhas zeros
The vertical asymptotes of r(x)are where the denominator has zeros
2
2
9( )
5 6
xr x
x x
Assignment
Lesson 4.6 Page 319 Exercises 1 – 41 EOO
93, 95, 99
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